[Ref in PA]

If A1, dribbling, stands perfectly still and bats the ball up in the air and then catches it, all without moving the feet, are you going to call that an illegal dribble??? I think not. The double touch rule is for when the dribble is continued and the ball touches the floor. Catching the ball ends the dribble, therefore the double touch rule does not apply. No where in the rules are the movement of feet considered as part of dribbling (except in case 4.15.4 E(b) where a dribble is being started), in fact, Note 1 at the end of 4-15-4 explicitly states that is is not possible to travel during a dribble.

The batting of the ball into the air does not seem to fit any of the definitions ending a dribble as outlined in 4-15-4.

The only way the OP play could be considered a violation is if you consider the dribble ending because of the bat of the ball. Somehow you must judge that the ball came to rest, thus ending the dribble. The movement and subsequent touch (catch and shot) would then be ruled an illegal dribble as in 4.15.4 E(b).

I do not consider the bat as described in the OP as the ball coming to rest, thus ending the dribble. Therefore, I consider the OP a legal play.